Writing in to cricinfo, Tim Parsons raises the oft repeated point that England were unfairly disadvantaged in the World T20 Cup by the application of the Duckworth Lewis system. The circumstances of that game are no longer terribly important, but the likelihood of further D/L problems remains.

Unfortunately, neither Tim, nor the commentors make a mathematical claim for why D/L is failing, except in so far as Tim claims - without evidence - that the small deduction in power play overs has advantaged the chasing team, and that there is limited data for T20 matches. This is a pity, because neither reason is correct, as far as I can tell, but there is a mathematical problem with the application of D/L in very short games.

The way D/L appears to be constructed (caveats for differences between standard and the proprietary professional edition), is to create a symmetrical target. That is, if a side is chasing, there is a certain point they need to be at, given their available overs and wickets, to win if the game is abandoned. the symmetry comes about because it is assumed, that if the batting side loses overs, that they lose resources in proportion to the overs lost. This produces a line of expectation, for the chasing side: the dark blue line in the graph below.

A short but relevant statistical aside

It is assumed, and even commented on at the link above, that the longer the period a team has to score at the set run-rate, the more difficult it is for them to achieve that. Which is true, empirically speaking, but not statistically.

Consider a situation where a team has unlimited batsmen and has to score at 8 an over to win. If the mean runs scored per over in this situation is greater than 8 then the more overs the team has, the better off they will be, because, probabilistically, the more overs there are, the more likely their run-rate will approach the mean (and the mean is above the required rate).

But, teams do not have unlimited batsmen. Even in a T20 game, their approach must be tempered against the need to preserve wickets. More importantly, even in a T20 game, the bowling side must take sufficient wickets to get batsmen to the crease for whom the mean scoring rate is below the required rate (tail-enders, in other words).

In which the problem with D/L becomes clear (ish)

Wicket-taking, therefore, becomes the imperative of the bowling side. And therein lies the problem with the resource reduction methods of D/L. Because the D/L is symmetrical, the reduction of several overs at the beginning of the innings assumes that the bowling side has had the worst possible start to the innings: no wickets.

Under the standard system, which projected that teams would chase the majority of the runs in a finishing burst, the chasing team was advantaged because the target for most of the game was below what they might reasonably need to be at. Under the professional system , specifically for higher run-rates, it was recognised that teams must chase a total almost linearly (or, to put it another way, the closer a required rate gets to the mean maximum scoring rate, the less conservative a team can afford to be). But an almost linear reduction for lost overs at the beginning of an innings results in a batting side getting their total reduced without any reduction in their ability to hit out.

Intuitively, this is recognised by those who argue a batting side should have their wicket allocation reduced in this situation, but there is a simpler method. Read backwards, the D/L resource table tells you the number of runs lost, with a reduction in the wicket resource (normally around 1-4). The simplest solution therefore, is to project how many wickets a bowling side might reasonably have taken in the lost overs, and add that number of runs to the required total. The effect on the curve can be seen in the light blue line, below.

For England, who might reasonably have expected to take 3-4 wickets in 11 overs of a T20 against the West Indies, the target would be adjusted from 81 to 87 or 88. Or roughly where most observers seem to think the target should have been. In a 50 over game, where wickets are less frequent, this sort of change would be less noticeable, which is, no doubt, why it has not been more obvious that over reductions favour the batting side slightly.